nLab
semimodel category

Contents

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

Introduced by Hovey 1998, the notion of semimodel categories s a relaxation of that of model categories which allows for a largely similar theory.

The notion of a weak model category and premodel category relaxes the definition even further.

Definition

(See Hovey 98, Theorem 3.3.)

A left semimodel category is a relative category equipped with a class of cofibrations and fibrations such that weak equivalences are closed under retracts and the 2-out-of-3 property, cofibrations have a left lifting property with respect to trivial fibrations, trivial cofibrations with cofibrant source have a left lifting property with respect to fibrations, and morphisms with cofibrant source can be factored as a cofibration followed by a fibration, either one of which can be further made trivial.

A right semimodel category is defined by passing to the opposite category.

Examples

Example

(semimodel structure on semisimplicial sets) There exists a right semi-model structure on the category of semi-simplicial sets (Rooduijn 2018).

References

The definition is due to:

The example of the semimodel structure on semisimplicial sets:

  • Jan Rooduijn, A right semimodel structure on semisimplicial sets, Amsterdam 2018 (pdf, mol:4787)

Last revised on June 26, 2021 at 08:00:52. See the history of this page for a list of all contributions to it.